Is there something like Layout.template_fileselector?
It's not possible to use a path variable from inside the Panel to achieve this effect. The fact that the used path is either used per object or per scene on the other hand offers a great workaround:

In your __init__.py-file you need the following code:

from bpy.props import StringProperty

def register():
    bpy.utils.register_module(__name__)

    # assigns a propery someName to every object. 
    # If you want one var per scene use bpy.types.Scene
    bpy.types.Object.someName = StringProperty(name="Exported Layer", 
        description="The UVLayer that will get exported",
        subtype='DIR_PATH') # so blender notices that as
                            # a directory path

if __name__ == "__main__":
    register()


Now you can use that variable inside yor panel:

from bpy.types import Panel

class MyExporter(Panel):
    """
    This exports the current object
    """

    bl_label = "Mobster Hunter"
    bl_space_type = "PROPERTIES"
    bl_region_type = "WINDOW"
    bl_context = "object"

    def draw(self, context):
        layout = self.layout
        col = layout.column()

        # use either context.object or context.scene and 
        # the same name
        col.prop(context.object, 'someName', text='Export directory')